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Re: Simplify with NestedLessLess?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106612] Re: Simplify with NestedLessLess?
  • From: Richard Fateman <fateman at cs.berkeley.edu>
  • Date: Mon, 18 Jan 2010 02:36:14 -0500 (EST)
  • References: <201001141049.FAA19892@smc.vnet.net> <4B4F39E7.1070002@wolfram.com> <4B4FAC81.7000108@ieee.org> <4B4FB26F.7050702@wolfram.com> <201001150821.DAA29881@smc.vnet.net> <op.u6k8eysrtgfoz2@bobbys-imac.local> <hiuuso$934$1@smc.vnet.net>

Dave Bird wrote:
> Interesting! But, I don't think I am correctly communicating what I'm 
> after yet. (Although, I admit that I am struggling some to keep up with 
> you guys in your Mathematica replies due to my inexperience.)
> 
> The original expression that I put up for illustration is:
> 
> -Cf^2 L2^2 Rg^2 Vg^4+3 (4 C Rg^2 Vd^2+4 Cf Rg^2 Vd^2+2 C Rg^2 Vd Vg)
> 
> We compare 4 C Rg^2 Vd^2 to 4 Cf Rg^2 Vd^2 because the two terms share 
> common coefficients so that they "reduce" to (4 Rg^2 Vd^2+4 Rg^2 Vd^2) 
> (C+Cf) . Thus it becomes obvious that C may be discarded w.r.t. Cf.
> 
> Please forgive if I have missed the correct application of your 
> suggestion, and thanks for the interest.
> 
> Dave
> 

Here's the deal.  You need to come up with a ranking of all the 
quantities in the expression in terms of some small parameter, say e 
[for epsilon]  You seem to know that C is smaller than Cf, from some
external facts.  Rewrite C   as C*e.
You seem to know that Cf is not one of those smallish guys, so you leave 
that alone.

All the other variables are either small, like C, or they are relatively 
large like Cf,  or they might even be VERY large, like 1/e .  If Q is 
very large, then Q*C is of the same size as a regular guy, like Cf. 
That means that (Cf+C*Q)  cannot be "simplified" to just Cf.

Anyway, to get back to the problem, you rephrase all of these variables 
in terms of e, and you expand in a series in e

(big mess)*e^0  + (big mess 2)*e^1 + ....

Depending on how much detail you want to retain, you can set e^1 to zero  or
e^2 to zero... and of course drop all higher order terms in e.

Your idea of kind of factoring and grouping and chopping things is 
impossible to use in a computer unless you specify which things are 
large and which are small.   That is, say we get a grouping of
(big mess)* (a +b).   Can a or b be dropped?  The computer cannot tell 
unless you say which things a, or b, or neither,  are large or small.

So you have to specify large or small (or very large etc.)

Once you have done that, the computer program is easy.  You use Series.

There is a tradition among computer algebra systems to have a kind of 
automatic system allowing people to specify "weights" on variables and 
do this style of calculation implicitly  without introducing 
specifically a new variable, e.g. epsilon.  Mathematica has not imitated 
this facility, at least as far as I know, yet. If someone wants details 
of these designs, I can respond off this list, since SteveC censors 
comparisons.

RJF


RJF


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